Estimation of a waveform period

ABSTRACT

Disclosed herein are systems and methods for estimating a period and frequency of a waveform. In one embodiment a system may comprise an input configured to receive a signal comprising a representation of the waveform. A period determination subsystem may calculate an estimated period of the signal based on a period determination function. An estimated period adjustment subsystem may determine an adjustment to the estimated period based on a result of the period determination function. A quality indicator subsystem configured to evaluate a measurement quality indictor function based on the estimated period, and to selectively update the period of the waveform based on the measurement quality indicator. A control action subsystem configured to implement a control action based on the period of the waveform.

RELATED APPLICATION

The present application claims priority under 35 U.S.C. §119(e) to U.S.Provisional Patent Application No. 62/120,467, filed Feb. 25, 2015, andtitled “Estimation of a Waveform Period,” which is incorporated hereinby reference in its entirety.

TECHNICAL FIELD

This disclosure relates to systems and methods for estimation of awaveform period and frequency. More particularly, this disclosurerelates to calculating an estimate of a period of a waveform as a timeshift that either maximizes a function of the waveform and the waveformafter the time shift, or determines a zero crossing of a function of awaveform's derivative and the waveform after the time shift.

BRIEF DESCRIPTION OF THE DRAWINGS

Non-limiting and non-exhaustive embodiments of the disclosure aredescribed, including various embodiments of the disclosure withreference to the figures included in the detailed description.

FIG. 1 illustrates a simplified one-line diagram of an electric powertransmission and distribution system consistent with embodiments of thepresent disclosure.

FIG. 2A illustrates a plot over time of a 60 Hz waveform with harmonicsin a steady state consistent with embodiments of the present disclosure.

FIG. 2B illustrates one period the waveform of FIG. 2A plotted against asampling rate of 10 kHz, together with a first version of the waveformsampled at a first sampling rate consistent with one embodiment of thepresent disclosure.

FIG. 2C illustrates one period the waveform of FIG. 2A plotted against asampling rate of 10 kHz, together with a second version of the signalsampled at a second sampling rate consistent with one embodiment of thepresent disclosure.

FIG. 2D illustrates a function representing a quality indicator over arange of time shift values, represented in terms of samples at 10 kHz,of the waveform illustrated in FIG. 2A.

FIG. 3A illustrates a plot over time of a waveform including harmonicshaving a time-varying frequency and a time-varying magnitude consistentwith embodiments of the present disclosure.

FIG. 3B illustrates an actual and a measured change in frequency overtime of the waveform of FIG. 3A determined using a data selection windowbetween t and t−2*T consistent with embodiments of the presentdisclosure.

FIG. 3C illustrates an actual and a measured change in frequency overtime of the waveform of FIG. 3A determined using a data selection windowbetween t−T and t+T consistent with embodiments of the presentdisclosure.

FIG. 3D illustrates one period the waveform of FIG. 3A plotted against asampling rate of 10 kHz, together with a first version of the waveformsampled at a first sampling rate consistent with one embodiment of thepresent disclosure.

FIG. 3E illustrates one period the waveform of FIG. 3A plotted against asampling rate of 10 kHz, together with a second version of the signalsampled at a second sampling rate consistent with one embodiment of thepresent disclosure.

FIG. 3F illustrates a function representing a quality indicator over arange of time shift values, represented in terms of samples at 10 kHz,of the waveform illustrated in FIG. 3A consistent with embodiments ofthe present disclosure.

FIG. 4A illustrates a plot over time of a waveform including harmonicshaving a time-varying frequency, a time-varying magnitude, and a phasejump consistent with embodiments of the present disclosure.

FIG. 4B illustrates one period the waveform of FIG. 4A plotted against asampling rate of 10 kHz, together with a first version of the waveformsampled at a first sampling rate consistent with one embodiment of thepresent disclosure.

FIG. 4C illustrates one period the waveform of FIG. 4A plotted against asampling rate of 10 kHz, together with a second version of the signalsampled at a second sampling rate consistent with one embodiment of thepresent disclosure.

FIG. 4D illustrates a function representing a quality indicator over arange of time shift values, represented in terms of samples at 10 kHz,of the waveform illustrated in FIG. 4A.

FIG. 5 illustrates a block diagram of a system configured to determinethe frequency and an indicator of periodicity of an input signalconsistent with embodiments of the present disclosure.

FIG. 6A illustrates plots over time of a heavily distorted waveform,together with a plot of frequency and magnitude changing over timeconsistent with embodiments of the present disclosure.

FIG. 6B illustrates a plot over time of the actual period, the rawperiod interpolated between samples, and the final period estimate afterlow-pass filtering, for a portion of the time illustrated in FIG. 6A.

FIG. 6C illustrates a plot over time of the actual and measuredfrequencies over time consistent with embodiments of the presentdisclosure.

FIG. 6D illustrates a comparison between an actual frequency of theinput signal, which is also illustrated in FIG. 6A, and a response of afrequency measurement system consistent with embodiments of the presentdisclosure.

FIG. 7 illustrates plots over time of an input voltage waveformincluding harmonics and a switching event, an actual and a measuredfrequency associated with the input signal, and a quality-of-measurementvalue consistent with embodiments of the present disclosure.

FIG. 8 illustrates plots over time of an input voltage waveform from alow-inertia generator showing subsynchronous oscillations and afast-frequency ramp after the generator becomes electrically islandedafter clearing a fault, a frequency measurement consistent withembodiments of the present disclosure, frequency measurement based onthe common zero-crossings method, and a quality-of-measurement valueconsistent with embodiments of the present disclosure.

FIG. 9 illustrates a flowchart of a method for estimating the period ofa waveform consistent with embodiments of the present disclosure.

FIG. 10 illustrates a functional block diagram of a system configured toestimate the period of a waveform consistent with embodiments of thepresent disclosure.

DETAILED DESCRIPTION

The present disclosure pertains to systems and methods for identifying aperiod of a periodic (i.e. repeating) waveform. Determination of thefrequency of a waveform may be applied in various applications. Inspecific application, the systems and methods disclosed herein may beapplied to an electrical power system. Analysis of the frequency of anelectric power system may be applied to frequency control and frequencyprotection, including load shedding and overexcitation protection.Frequency determination may also be used in synchrophasor measurementsand in protective relays and meters for frequency tracking in order tokeep alternating current measurements, especially phasors, accurate.

As used herein, a period is a time shift that may be applied to thewaveform in order for it to repeat. Real-world signals, such as thefrequency of an electrical power system, are typically in a state ofconstant fluctuation. Such fluctuations in an electric power system maybe created as generators oscillate around equilibrium points in a stablepower system and fluctuations in the loads placed on such generators.Major system events can cause pronounced oscillations as generatorsspeed up and slow down with respect to the rest of the power system.Some network configurations may lead to resonant conditions causingsubsynchronous oscillations. As a result, power system voltages andcurrents are never in a perfect steady state and, therefore, are notprecisely periodic.

Systems and methods consistent with the present disclosure may utilizean autocorrelation of the waveform or correlation of the waveform andits derivative using a freely chosen window length. One period of thewaveform is a logical choice for the length of the correlation window. Awindow of half of a period is insufficient because the input waveform ingeneral does not have to be anti-symmetrical with respect to themid-point of the period (i.e., x(t−T/2) does not have to equal −x(t)).Using multiple periods adds latency, and at least in theory, it does notcontribute additional information. The exact one period is the minimumrequired and the logical choice; however, other considerations may alsobe utilized to select an autocorrelation window.

, a period of a periodic waveform is the time shift T₀ for which anintegral over a time period T₀ of the product of the waveform and thewaveform shifted by T₀ is at its maximum. In other words:

A(t,T)=∫_(t-T) ^(t) x(t)·x(t−T)dt  Eq. 1A

T ₀(t)=T for which A(t,T) is at its maximum  Eq. 1B

As the waveform is shifted by T and integrated over T, the value ofintegral may increase with T even if T is different than the preciseperiod of the waveform. The increase is not due to improved correlation,but because the function is integrated over a longer period of time.Accordingly, in various embodiments, the value of Eq. 1 may benormalized based on the length of the window.

The period of a periodic (i.e. repeating) waveform may be determinedusing Eq. 2A and 2B. In Eq. 2A and 2B is the time shift T₀ for which anintegral over a time period T₀ of the product of the waveform and thewaveform shifted by T₀, normalized with the wave magnitude, is at itsmaximum.

$\begin{matrix}{{A\left( {t,T} \right)} = \frac{\int_{t - T}^{t}{{{x(t)} \cdot {x\left( {t - T} \right)}}\ {t}}}{\int_{t - {2T}}^{t}{{{x(t)} \cdot {x(t)}}\ {t}}}} & {{{Eq}.\mspace{14mu} 2}A} \\{{T_{0}(t)} = {T\mspace{14mu} {for}\mspace{14mu} {which}\mspace{14mu} {A\left( {t,T} \right)}\mspace{14mu} {is}\mspace{14mu} {at}\mspace{14mu} {its}\mspace{14mu} {maximum}}} & {{{Eq}.\mspace{14mu} 2}B}\end{matrix}$

For a periodic waveform A(t,T₀) the value of Eq. 2A is 1. For anear-periodic waveform such as during a frequency ramp and/or amagnitude ramp or oscillation the A(t,T₀) value of Eq. 2A is below 1,but close to 1. For a non-periodic waveform such as for a phase ormagnitude jump the value of Eq. 2A is considerably below 1. As a result,in various embodiments the value of Eq. 2A may provide a qualityindicator reflecting the periodicity of the waveform. A waveform may beconsidered periodic if A(t,T₀) exceeds a specific threshold. In onespecific embodiment, a waveform may be considered periodic ifA(t,T₀)>0.95. Other embodiments may vary the threshold on variousfactors, including rate of change of a signal, the magnitude of thefrequency, or other parameters of the signal. An evaluation of theperiodicity of a signal may be determined at any arbitrary point intime, t. In one embodiment, a sliding data window of two waveformperiods may be used to evaluate Eq. 2A.

Eq. 2 may be well suited for off-line applications or embodiments havingample computing power. In other embodiments, the latency of a systemconsistent with the present disclosure may be improved or thecomputational requirements may be reduced by using a morecomputationally efficient algorithm. Solving Eq. 2 to find a maximumcontributes to the computational complexity. Accordingly, in alternativeembodiments, a zero-crossing function may be used rather thanidentifying a maximum. Finding the zero-crossing may be numericallysimpler, and thus may decrease latency or decrease the computationalrequirements or latency of a system consistent with the presentdisclosure.

A periodic signal may include multiple components, including afundamental frequency and a number of harmonics components. The harmoniccomponents are multiples of the fundamental frequency. For a sinewavesignal (or a component of a sinewave sign), the signal and its timederivative are orthogonal, i.e. are shifted by 90 electrical degrees atthe frequency of the said signal. As a result of this orthogonality, thesum of the signal and its derivative taken over the period of the saidsignal is zero. Moreover, the sum over any number of periods is zero. Inaddition a sum of the product of two different harmonics over a periodof the lower harmonic is zero. Accordingly, as expressed in Eqs. 3A and3B, a period of a periodic (i.e. repeating) waveform T₀ corresponds tothe length of a integration window for which the integral of the productof the signal x(t−T) shifted by T₀ and its time derivative, x′, is zero.In other words:

B(t,T)=∫_(t-T) ^(t) x(t−T)·x′(t)dt  Eq. 3A

T ₀(t)=T for which B(t,T) is at zero  Eq. 3B

Alternatively, the shift may be applied to the signal derivative insteadof the signal, as expressed in Eq. 3C.

B(t,T)=∫_(t-T) ^(t) x′(t−T)·x(t)dt  Eq. 3C

Solving Eq. 3 to find a zero may provide a numerically simplercalculation in comparison to solving Eq. 2A to find a maximum. Asdescribed above, Eq. 2A may also be used to assess the periodicity of awaveform, and thus, to assess the quality of a resulting determinationof the determined period of the waveform. In cases where the waveform isnot periodic, the determined period may be of limited use. In someembodiments, Eq. 3 may be used to determine the period of waveform, andEq. 2 may be used to assess the quality of the determination of theperiod. In other words, a period determined using Eq. 3A, may serve asan input to Eq. 2A, which may provide an indication of whether thesignal is truly periodic. Eq. 2 may be referred to in certainembodiments as a periodicity determination function, and Eq. 3 may bereferred to as a period determination function. Of course, otherequations or representations may also be used to determine the periodand periodicity of a waveform consistent with the present disclosure.

When Eq. 3A is calculated for a period T that is shorter than the actualperiod To, the function per Eq. 3A is positive, and when calculated fora period T that is longer than the actual period To, the function isnegative. Accordingly, if the value of Eq. 3A is positive a presentperiod estimate may be increased. Similarly, if the value of Eq. 3A isnegative a present period estimate may be decreased. Further, the valueof Eq. 3A does not change substantially when calculated for twoconsecutive samples. As a result, a system implementing Eq. 3A maycalculate the equation once for any given point in time, and the systemmay increase or decrease the period estimate for the calculation of theperiod determination function for the next data sample. If the samplingfrequency is sufficiently high (e.g. 4 kHz or higher), the process isstable even for high rates-of-change of frequency. The value of theperiod (i.e., the shift in the signal) is not limited to integermultiples of samples. In some embodiments, interpolation may be used todetermine non-integer multiples of samples while maintainingcomputational efficiency.

Eq. 2 effectively looks at the data window between t and t−2*T. Certainembodiments consistent with the present disclosure may tag themeasurement with respect to the middle of an effective data window, asshown in Eq. 4A-4B.

$\begin{matrix}{{A\left( {t,T} \right)} = \frac{\int_{t}^{t + T}{{{x(t)} \cdot {x\left( {t - T} \right)}}\ {t}}}{\int_{t - T}^{t + T}{{{x(t)} \cdot {x(t)}}\ {t}}}} & {{{Eq}.\mspace{14mu} 4}A} \\{{T_{0}(t)} = {T\mspace{14mu} {for}\mspace{14mu} {which}\mspace{14mu} {A\left( {t,T} \right)}\mspace{14mu} {is}\mspace{14mu} {at}\mspace{14mu} {its}\mspace{14mu} {maximum}}} & {{{Eq}.\mspace{14mu} 4}B}\end{matrix}$

Stated in other words, Eq. 4A may use the effective data window betweent−T and t+T. Use of the effective data window between t−T and t+T may bewell-suited for embodiments using synchrophasor frequency measurementbecause it introduces no group delay. In various embodiments, Eq. 4A-4Bmay be used as a definition of period/frequency for a power systemsignal is a quasi-steady state. The value of A(t,T) corresponding to themeasured period may be communicated or checked locally and may assert aflag if the wave is periodic at the time.

In various embodiments, synchrophasors consistent with the C37.118standard, may be used to represent measurements used in variousembodiments of the present disclosure. A phasor may be defined as a sinewave that best estimates the periodic wave. In some embodiments, aleast-error-squared fit algorithm may be used to determine the phasorparameters, which is the same as the fundamental frequency harmonic ofthe Fourier transform. Accordingly, an instantaneous phasor is a“frequency tracked” one-cycle Fourier. The phasor and period (frequency)may be calculated on the same data in various embodiments, and mayconstitute a coherent pair with the Fourier-based phasor by the virtueof using the same data points.

The foregoing concepts may be applied in various applications todetermine the period and periodicity of time-varying signals. Onespecific application involves analyzing alternating current signals inan electric power distribution system. Variations in the frequency of analternating current in an electric power distribution system provideinformation regarding the performance of the system. Such informationmay be used to control the electric power distribution system andenhance the stability or the reliability of the system.

Eq. 4A may define a period (frequency) at the time tag t, using theobservations window t−T to t+T. This window may be independent from areporting rate. Actual Phasor Measurement Units (PMUs) can implement anymethod to measure frequency consistent with the present disclosure. Eq.4A defines a consistent reference for any arbitrary signal. One cycleFourier defines what a phasor is. This is calculated at t using the datawindow of t−T/2 to t+T/2 (one cycle Fourier) or t−T to t+T (two-cycleFourier), etc. The phasor and frequency pair is a coherentcharacterization of a period waveform at the point in time t. The time,t, is referenced to absolute time, and the angle of the phasor isreferenced accordingly by placing the cosine and sine windowsaccordingly to the time tag. One cycle data window may facilitatecommunication of measured phasors at high rates. Where the data rate isnot high enough to satisfy the sampling theorem, an anti-aliasing filtermay be applied to the phasor and frequency.

The embodiments of the disclosure will be best understood by referenceto the drawings. It will be readily understood that the components ofthe disclosed embodiments, as generally described and illustrated in thefigures herein, could be arranged and designed in a wide variety ofdifferent configurations. Thus, the following detailed description ofthe embodiments of the systems and methods of the disclosure is notintended to limit the scope of the disclosure, as claimed, but is merelyrepresentative of possible embodiments of the disclosure. In addition,the steps of a method do not necessarily need to be executed in anyspecific order, or even sequentially, nor need the steps be executedonly once, unless otherwise specified.

In some cases, well-known features, structures or operations are notshown or described in detail. Furthermore, the described features,structures, or operations may be combined in any suitable manner in oneor more embodiments. It will also be readily understood that thecomponents of the embodiments as generally described and illustrated inthe figures herein could be arranged and designed in a wide variety ofdifferent configurations.

Several aspects of the embodiments described may be implemented assoftware modules or components. As used herein, a software module orcomponent may include any type of computer instruction or computerexecutable code located within a memory device and/or transmitted aselectronic signals over a system bus or wired or wireless network. Asoftware module or component may, for instance, comprise one or morephysical or logical blocks of computer instructions, which may beorganized as a routine, program, object, component, data structure,etc., that performs one or more tasks or implements particular abstractdata types.

In certain embodiments, a particular software module or component maycomprise disparate instructions stored in different locations of amemory device, which together implement the described functionality ofthe module. Indeed, a module or component may comprise a singleinstruction or many instructions, and may be distributed over severaldifferent code segments, among different programs, and across severalmemory devices. Some embodiments may be practiced in a distributedcomputing environment where tasks are performed by a remote processingdevice linked through a communications network. In a distributedcomputing environment, software modules or components may be located inlocal and/or remote memory storage devices. In addition, data being tiedor rendered together in a database record may be resident in the samememory device, or across several memory devices, and may be linkedtogether in fields of a record in a database across a network.

Embodiments may be provided as a computer program product including anon-transitory computer and/or machine-readable medium having storedthereon instructions that may be used to program a computer (or otherelectronic device) to perform processes described herein. For example, anon-transitory computer-readable medium may store instructions that,when executed by a processor of a computer system, cause the processorto perform certain methods disclosed herein. The non-transitorycomputer-readable medium may include, but is not limited to, harddrives, floppy diskettes, optical disks, CD-ROMs, DVD-ROMs, ROMs, RAMs,EPROMs, EEPROMs, magnetic or optical cards, solid-state memory devices,or other types of machine-readable media suitable for storing electronicand/or processor executable instructions.

FIG. 1 illustrates a simplified one-line diagram of an alternatingcurrent electric power transmission and distribution system 100consistent with embodiments of the present disclosure. Electric powerdelivery system 100 may be configured to generate, transmit, anddistribute electric energy to loads. Electric power delivery systems mayinclude equipment, such as electric generators (e.g., generators 110,112, 114, and 116), power transformers (e.g., transformers 117, 120,122, 130, 142, 144 and 150), power transmission and delivery lines(e.g., lines 124, 134, 136, and 158), circuit breakers (e.g., breakers152, 160, 176), busses (e.g., busses 118, 126, 132, and 148), loads(e.g., loads 140, and 138) and the like. A variety of other types ofequipment may also be included in electric power delivery system 100,such as voltage regulators, capacitor banks, and a variety of othertypes of equipment.

Substation 119 may include a generator 114, which may be a distributedgenerator, and which may be connected to bus 126 through step-uptransformer 117. Bus 126 may be connected to a distribution bus 132 viaa step-down transformer 130. Various distribution lines 136 and 134 maybe connected to distribution bus 132. Distribution line 136 may lead tosubstation 141 where the line is monitored and/or controlled using IED106, which may selectively open and close breaker 152. Load 140 may befed from distribution line 136. Further step-down transformer 144 incommunication with distribution bus 132 via distribution line 136 may beused to step down a voltage for consumption by load 140.

Distribution line 134 may lead to substation 151, and deliver electricpower to bus 148. Bus 148 may also receive electric power fromdistributed generator 116 via transformer 150. Distribution line 158 maydeliver electric power from bus 148 to load 138, and may include furtherstep-down transformer 142. Circuit breaker 160 may be used toselectively connect bus 148 to distribution line 134. IED 108 may beused to monitor and/or control circuit breaker 160 as well asdistribution line 158.

Electric power delivery system 100 may be monitored, controlled,automated, and/or protected using intelligent electronic devices (IEDs),such as IEDs 104, 106, 108, 115, and 170, and a central monitoringsystem 172. In general, IEDs in an electric power generation andtransmission system may be used for protection, control, automation,and/or monitoring of equipment in the system. For example, IEDs may beused to monitor equipment of many types, including electric transmissionlines, electric distribution lines, current transformers, busses,switches, circuit breakers, reclosers, transformers, autotransformers,tap changers, voltage regulators, capacitor banks, generators, motors,pumps, compressors, valves, and a variety of other types of monitoredequipment.

As used herein, an IED (such as IEDs 104, 106, 108, 115, and 170) mayrefer to any microprocessor-based device that monitors, controls,automates, and/or protects monitored equipment within system 100. Suchdevices may include, for example, remote terminal units, differentialrelays, distance relays, directional relays, feeder relays, overcurrentrelays, voltage regulator controls, voltage relays, breaker failurerelays, generator relays, motor relays, automation controllers, baycontrollers, meters, recloser controls, communications processors,computing platforms, programmable logic controllers (PLCs), programmableautomation controllers, input and output modules, and the like. The termIED may be used to describe an individual IED or a system comprisingmultiple IEDs.

A common time signal may be distributed throughout system 100. Utilizinga common or universal time source may ensure that IEDs have asynchronized time signal that can be used to generate time synchronizeddata, such as synchrophasors. In various embodiments, IEDs 104, 106,108, 115, and 170 may receive a common time signal 168. The time signalmay be distributed in system 100 using a communications network 162 orusing a common time source, such as a Global Navigation Satellite System(“GNSS”), or the like.

According to various embodiments, central monitoring system 172 maycomprise one or more of a variety of types of systems. For example,central monitoring system 172 may include a supervisory control and dataacquisition (SCADA) system and/or a wide area control and situationalawareness (WACSA) system. A central IED 170 may be in communication withIEDs 104, 106, 108, and 115. IEDs 104, 106, 108 and 115 may be remotefrom the central IED 170, and may communicate over various media such asa direct communication from IED 106 or over a wide-area communicationsnetwork 162. According to various embodiments, certain IEDs may be indirect communication with other IEDs (e.g., IED 104 is in directcommunication with central IED 170) or may be in communication via acommunication network 162 (e.g., IED 108 is in communication withcentral IED 170 via communication network 162).

Communication via network 162 may be facilitated by networking devicesincluding, but not limited to, multiplexers, routers, hubs, gateways,firewalls, and switches. In some embodiments, IEDs and network devicesmay comprise physically distinct devices. In other embodiments, IEDs andnetwork devices may be composite devices, or may be configured in avariety of ways to perform overlapping functions. IEDs and networkdevices may comprise multi-function hardware (e.g., processors,computer-readable storage media, communications interfaces, etc.) thatcan be utilized in order to perform a variety of tasks that pertain tonetwork communications and/or to operation of equipment within system100.

In various embodiments, IEDs 104, 106, 108, 115, and 170 may beconfigured to monitor the frequency of alternating current waveforms insystem 100. The IEDs may utilize common time source 168 to time-alignmeasurements for comparison across system 100. The measurements may beused to in connection with the systems and methods disclosed herein forcontrol of system 100.

In various embodiments, a signal in system 100 may be representative ofan operating condition of the electric power delivery system. In oneembodiment, the signal may be a current signal obtained from theelectric power delivery system using a current transformer or the like.A secondary of the current transformer may be in electricalcommunication with an IED configured to monitor the electric powerdelivery system, and which further may be configured to provideprotection to the electric power delivery system. The IED may includefurther current transformers in electrical communication with theprimary current transformer. The IED may also include ananalog-to-digital (A/D) converter in communication with the currenttransformer to digitize samples of the obtained waveform. The IED mayinclude a processor receiving such digitized samples (which may undergofurther processing using, for example, filters and the like) and usingthe embodiments described herein, the IED may calculate a period of thewaveform therefrom. In other embodiments, the waveform may berepresentative of a voltage of the electric power delivery system.

FIG. 2A illustrates a plot over time of a 60 Hz waveform 200 heavilydistorted with harmonics in a steady state consistent with embodimentsof the present disclosure. FIGS. 2A-2D illustrate an example of aprocess that may be used in various embodiments for determining a periodof the waveform illustrated in FIG. 2A using techniques consistent withthe present disclosure. The waveform illustrated in FIG. 2A is adistorted sinusoidal wave with a frequency of 60 Hz. In a system sampledat 10 kHz, the waveform would have a period of 166.667 samples. Inconnection with FIGS. 2B-2D, a sampling rate of 10 kHz is used, althoughother sampling rates could be utilized in connection with variousembodiments.

In various embodiments, Eq. 2 may be implemented directly by replacingthe continuous time integrals with sums of a plurality of samples.Interpolation can be used to find the maximum of Eq. 2 between twointeger samples of T. Accordingly, the system may avoid use of anotherinterpolation to delay the signal by exactly one period (including thefractional period), and the system may avoid another interpolation torun the sums over an exact period (including the fractional period).Instead, the system may run calculations for values of a periodexpressed in integer number of samples. A system configured to determinethe period of waveform 200 may begin with an initial estimate of theperiod being the period measured in the previous measurement cycle.

FIG. 2B illustrates one period the waveform 200 of FIG. 2A plottedagainst a sampling rate of 10 kHz, together with a first version of thewaveform 202 shifted by a first value consistent with one embodiment ofthe present disclosure. The first time shift may represent the initialestimate of the period. In the illustrated example, the first timeshift, or initial estimate, is 164 samples. The system may calculate Eq.3A for T=164 samples and may determine that the result is positive. Thepositive result indicates that the value T=164 is below the actualperiod. Accordingly, the first version of the waveform 202 lags behindwaveform 200. Eq. 2A may be evaluated for the scenario illustrated inFIG. 2B. At 164 samples per period, Eq. 2A yields a value of 0.9827. Thevalue of Eq. 2A is relatively close to 1 given that the first samplingrate is relatively close to the actual sampling rate.

FIG. 2C illustrates one period the waveform 200 of FIG. 2A plottedagainst a sampling rate of 10 kHz, together with a second version of thesignal 206 shifted by a second time shift consistent with one embodimentof the present disclosure. As noted in connection with FIG. 2B, thevalue T=164 is below the actual period, and accordingly the estimate maybe increased. In the illustrated example, the second time shift isincreased to T=170, which is greater than the actual period.Accordingly, the second version of the waveform 206 leads waveform 200,and the value of Eq. 3B is negative. At 170 samples per period, Eq. 2Ayields a value of 0.9743.

The change in the sign of the result of Eq. 3A provides an indicationthat the actual period is between T=164 and T=170. A simple algorithmlimited to the period estimate in integer sample counts would oscillatethe estimated period between 166 and 167 samples. Note that the straightaverage between 166 and 167 (166.5) does not reflect the true period of166.6(6). Resampling would allow for the application of fractionalshifts; however, resampling may utilize significant computationalresources. The computational efficiency may be increased by usingperiods expressed in integer sample counts and interpolating if theresult of Eq. 3A changes sign between two consecutive samples. In thisexample, interpolation between the two points (166 samples) and (167samples) indicates a period of 166.67 samples. Accordingly, an algorithmconsistent with the present disclosure may constantly track the zero ofEq. 3A. The interpolation allows for better accuracy. If the perioddetermination function does not change signs between two consecutivesamples, the algorithm uses the newest integer value of the period.

FIG. 2D illustrates a function 206 which represents a quality indicatorover a range of time shifts, represented in terms of samples at 10 kHz,of the waveform illustrated in FIG. 2A. As would be expected given thatthe signal is periodic and has a frequency of 60 Hz, function 206reaches a maximum of 1 at a 166.667 samples. In some embodiments, thevalue may be obtained by interpolation between the period values of 166and 167. The analysis illustrated in connection with FIGS. 2B-2D may beperformed for an arbitrary waveform having an unknown period and unknownperiodicity.

FIG. 3A illustrates a plot over time of a waveform 300 heavily distortedwith harmonics and having a time-varying frequency and a time-varyingmagnitude consistent with embodiments of the present disclosure. FIGS.3A-3E illustrate a process for determining a period of the waveformillustrated in FIG. 3A using techniques consistent with the presentdisclosure. The time-varying change of the magnitude is indicated byline 302. The waveform illustrated in FIG. 3A is a waveform having afrequency change of −15 Hz per second, as best shown in FIG. 3B, and amagnitude change of −80% of initial value per second.

FIG. 3B illustrates an actual and a measured change in frequency overtime of the waveform of FIG. 3A determined using a data selection windowbetween t and t−2*T consistent with embodiments of the presentdisclosure. The rate of change of the frequency is −15 Hz per second.Line 304 reflects an actual frequency, and line 306 reflects a measuredfrequency. The delay between line 304 and line 306 may be attributableto a data selection window between t and t−2*T.

FIG. 3C illustrates an actual and a measured change in frequency overtime of the waveform of FIG. 3A determined using a data selection windowbetween t−T and t+T consistent with embodiments of the presentdisclosure. As discussed above in connection with Eq. 4A, in certainembodiments, the effective data window may be used between t−T and t+T,rather than the data selection window between t and t−2*T used inconnection with Eq. 2A. As may be appreciated by comparing FIG. 3B toFIG. 3C, there is no delay in FIG. 3C between the actual frequency andthe measured frequency.

FIG. 3D illustrates one period the waveform 300 of FIG. 3A plottedagainst a sampling rate of 10 kHz, together with a first version of thewaveform 310 time shifted by a first value consistent with oneembodiment of the present disclosure. The first time shift, at 177samples per period, is slightly lower than the actual period of thewaveform 300. Accordingly, the first version of the waveform 310 lagsbehind waveform 300.

FIG. 3E illustrates one period the waveform 300 of FIG. 3A plottedagainst a sampling rate of 10 kHz, together with a second version of thewaveform 312 time shifted by a second value consistent with oneembodiment of the present disclosure. The second time shift, at 183samples per period, is slightly higher than the actual period ofwaveform 300. Accordingly, the second version of the waveform 312 leadswaveform 300.

FIG. 3F illustrates a function 314 representing a quality indicator Eq.2A over a range of time shifts, represented in terms of samples at 10kHz, of the waveform illustrated in FIG. 3A consistent with embodimentsof the present disclosure. The maximum equals 0.9997, signifying thewaveform is highly periodic, at 179.64 samples per period. In someembodiments, the maximum value may be obtained by interpolation betweencertain period values. The analysis illustrated in connection with FIGS.3D-3F may be performed for an arbitrary waveform having an unknownperiod and unknown periodicity.

FIG. 4A illustrates a plot over time of a waveform 400 heavily distortedwith harmonics, having a time-varying frequency, a time-varyingmagnitude, and a phase jump at a time designated by line 402. Thetime-varying frequency and time-varying magnitude of waveform 400 is thesame as waveform 300, with the exception of the 90° phase jump at t=0.3sec. Events that shift the angle of a waveform affects the periodicityof the waveform. Such events may include switching events in an electricpower distribution system. Switching events cause the zero crossings toshift instantly, creating very large frequency measurement errors whenusing the zero-crossing method. Such errors may be filtered out byrejecting wrong measurements rather than by averaging, hence the use ofvarious heuristic approaches to validating or rejecting such rawmeasurements.

Methods consistent with the present disclosure may run at a fixed rate,such as each time a new input waveform sample becomes available. A fixedschedule makes raw frequency post-filtering more straightforward. Aswith any frequency measuring method, the method behaves poorly fornonperiodic waveforms.

FIG. 4B illustrates one period the waveform 400 of FIG. 4A plottedagainst a sampling rate of 10 kHz, together with a first version of thewaveform 404 shifted by a first value consistent with one embodiment ofthe present disclosure. The first time shift is 177 samples per period.Following the phase jump at the time indicated by line 402, the secondwaveform 404 begins to deviate substantially from waveform 400.

FIG. 4C illustrates one period the waveform 400 of FIG. 4A plottedagainst a sampling rate of 10 kHz, together with a second version of thesignal 404 shifted by a second value consistent with one embodiment ofthe present disclosure. The second time shift, at 180 samples perperiod, closely correlates to waveform 400 prior to the phase jump atthe time indicated by line 402. Following the phase jump at the timeindicated by line 402, the second waveform 404 begins to deviatesubstantially from waveform 400.

FIG. 4D illustrates a function 406 representing a quality indicator Eq.2A over a range time shifts, represented in terms of samples at 10 kHz,of the waveform illustrated in FIG. 4A. The maximum equals 0.635,signifying the waveform is not periodic. In various embodiments, such athreshold may be applied to the value of Eq. 2A to determine whether awaveform is periodic. In one specific embodiment, the threshold may be0.95.

FIG. 5 illustrates a block diagram of a system configured to determinethe frequency and an indicator of periodicity of an input signalconsistent with embodiments of the present disclosure. System 500receives an input signal 502. In various embodiments, input signal 502may comprise a representation of an alternating voltage or current in anelectrical power system. The input signal may be an input to aderivative function 504 and a buffer 506. The output of the derivativefunction 504 and the buffer may be multiplied and provided as an inputto a period determination function 508. In various embodimentsconsistent with the present disclosure, period determination function508 may solve Eq. 3A and 3B.

The output of the period determination function 508 may be an input to aperiod estimate adjustment function 510. The period estimate adjustmentfunction 510 may be configured to increase or decrease an estimate ofthe period of the waveform based on the result of the perioddetermination function 508. In embodiments in which period determinationfunction 508 implements Eq. 3A, if the result is positive, the periodmay be increased in a successive iteration; if the result is negative,the period may be decreased in a successive iteration.

An interpolation function may be used to interpolate the value of theperiod. As described above, the output of the period determinationfunction 508 may be positive if the present period estimate is below theactual period, and negative if the present period estimate is above theactual period. In some embodiments, interpolation function 512 may beinterpolated between successive samples of the estimated period thatresult in a sign change in the output of period determination function508.

The output of the interpolation function 512 may be passed through a lowpass filter 514. Low pass filter 514 may increase the accuracy of thefrequency calculation obtained from interpolation. The lower the cut-offfrequency of low pass filter 514, the better the accuracy and the slowerthe response time of the algorithm (the lag between the true andmeasured frequency). Low-pass filter 514 may provide a simple andeffective way to control accuracy-versus-speed performance requirements.In various embodiments, low pass filter 514 may be set at about 15 Hz inorder to allow frequency changes as fast at 15 Hz/s. In one specificembodiment, two filters may be used: one for frequency metering purposesand the other for frequency protection or frequency tracking purposes.

In some embodiments, the output of system 500 may output the frequency,rather than the period of the waveform. In such embodiments, thefrequency is determined as the sampling frequency divided by the period.

A periodicity determination function 516 may be configured to determinethe extent to which the input signal 502 is periodic. In someembodiments, periodicity determination function 516 may determine theperiodicity of the input signal 502 using Eq. 2A. In some embodiments,the periodicity may reflect a numerical value reflecting the extent towhich the input signal 502 is periodic. In some embodiments, a value of1 may be used to quantify a completely periodic waveform. A value below,but close to 1 may reflect a nearly-periodic waveform, such as afrequency ramp, magnitude ramp, or oscillations in an electric powersystem. A value substantially below 1 may quantify a non-periodicwaveform. In some embodiments, a threshold value may be established todifferentiate between periodic and non-periodic waveforms.

The output of periodicity determination function 516 may also be passedthrough a low pass filter 518 to improve the accuracy of the periodicityoutput and to provide time coherency between the filtered period T1 andthe periodicity value.

FIG. 6A illustrates plots over time of a heavily distorted waveform,together with a plot of the frequency and magnitude changing over timeconsistent with embodiments of the present disclosure. As illustrated,the frequency declines at 20 Hz/s, exhibits a step change by 5 Hz, andlater increases at 2.5 Hz/s. The magnitude of the fundamental frequencybegins a decline at t=1 second. FIG. 6B shows the actual period, the rawperiod interpolated between samples, and the final period estimate afterlow-pass filtering, for a fraction of the test time. The measurementerror in the steady state is small, on the order of 0.6 mHz. FIG. 6Cshows the actual and measured frequencies and illustrates how well thenew algorithm tracks frequency despite magnitude changes and significantharmonic distortions in the input signal.

FIG. 6D illustrates a comparison between an actual input signal, whichis also illustrated in FIG. 6A, and a response of a system consistentwith embodiments of the present disclosure. As illustrated, the systemclosely tracks the input signal in spite of the distortions in thesignal.

FIG. 7 illustrates plots over time of an input voltage waveformincluding a switching event (phase jump by 90 degrees at t=0.3 s), anactual and a measured frequency associated with the input signal, and aquality-of-measurement value consistent with embodiments of the presentdisclosure. As illustrated, the frequency measurement is affected duringthe phase jump that occurs at t=0.3 s. Specifically, upon occurrence ofthe phase jump the measured frequency begins to increase. Afterincreasing for a period, the measured frequency begins to decrease andreturns to the initial value after a few cycles of the input waveform.

The quality-of-measurement index decreases as the error between themeasured and actual frequency values diverges. A threshold 702 maydifferentiate between valid and invalid frequency measurements. Asillustrated, following the phase jump, the quality-of-measurement indexvalue falls below threshold 702, and as such may trigger varioustechniques for addressing the disparity between the measured and theactual frequency values. In one specific embodiment, the last validfrequency measurement may be held until the quality-of-measurement indexexceeds a threshold 702.

The results illustrated in FIG. 7 include several desirable attributes.Specifically, the results show a quick recovery from the phase jump. Inthe illustrated example, the frequency measurement is marked as invalidfor about 40 ms, which may correspond to the time needed to pass throughthe data window used to determine the frequency value. In addition, themeasured frequency accurately tracks the frequency of the input waveformin spite of heavy distortion, including multiple zero crossings withineach fundamental frequency period.

FIG. 8 illustrates plots over time of an input voltage waveform from alow-inertia generator showing subsynchronous oscillations and afast-frequency ramp after the generator becomes electrically islandedafter clearing a fault, a frequency measurement using a zero-crossingsmethod, a frequency measurement consistent with the present disclosureand a quality-of-measurement value consistent with embodiments of thepresent disclosure. The illustration in FIG. 8 illustrates desirableperformance in comparison to the zero-crossing approach (modeled with noprefiltering). The quality-of-measurement index correctly identifiesfrequency measurements that are less accurate and which may be excludedfrom downstream applications of frequency.

In some embodiments, systems and methods consistent with the presentdisclosure may provide accuracy on the order of ten mHz or better.Typical system conditions may include frequency excursions of ±5 Hz andfrequency ramps of up to ±15 Hz/s. Inertia-free inverter-based powersources pose higher requirements than the traditional requirementslisted above. Various embodiments of the present disclosure may satisfysuch requirements.

FIG. 9 illustrates a flowchart of a method 900 for estimating the periodof a waveform consistent with embodiments of the present disclosure.Method 900 may incrementally determine the period of the waveform andthen track the period as new data is received. At 902, an initialfrequency estimate may be established. The estimate is not necessarilybased on information associated with the waveform. Rather, the initialestimate may simply be a starting point. In some embodiments, theinitial frequency estimate may be set at a nominal frequency for asystem. At 904, the value of a period determination function may becalculated based on the frequency estimate. In certain embodiments, theperiod determination may utilize Eq. 3. In such embodiments, the sign(i.e., whether the result is positive or negative) of the result of Eq.3 may provide an indication of whether the frequency estimate is aboveor below the actual frequency. At 906, the frequency estimate may beincrementally adjusted based on the result of the period determinationfunction. In some embodiments, the value may be incremented ordecremented in specific quantities (e.g., one sample). At 908, anupdated value of the period determination function may be calculatedbased on the updated frequency estimate.

At 910, method 900 may determine whether the actual period of thewaveform falls between the prior frequency estimate and the updatedfrequency estimate. The determination may be based, in some embodiments,on the result of the period determination function. In one specificembodiment in which Eq. 3 is used in connection with the perioddetermination function, the sign of the result may switch depending onwhether the updated frequency estimate is above or below the actualfrequency. As such, a change in the sign of the result of Eq. 3 maycause method 900 to diverge at 910. If the actual period is notdetermined to be between the prior frequency estimate and the currentfrequency estimate, method 900 may return to 906.

At 912, method 900 may interpolate the value of the actual period of thewaveform between the prior frequency estimate and the updated frequencyestimate. In various embodiments, Eq. 2 may be implemented directly byreplacing the continuous time integrals with sums of a plurality ofsamples. Interpolation can be used to find the maximum of Eq. 2A betweentwo integer samples of T. Accordingly, the system may use anotherinterpolation to delay the signal by exactly one period (including thefractional period), and the system may avoid another interpolation torun the sums over an exact period (including the fractional period).Instead, the system may run calculations for values of period expressedin integer number of samples.

At 914, a measurement quality indicator function may be evaluated. Invarious embodiments, Eq. 2 may provide an indication of the periodicityof the waveform, and thus, an indication of the quality of thedetermined period is valid. In some embodiments, at 915, a variable gainmay be applied based on the quality indicator function. The variablegain may be applied after a filtering process, in embodiments thatutilize a filter. The variable gain may increase the relative weight ofdata samples when the quality indicator function indicates that theinput function is highly periodic, and to decrease the relative weightof data samples when the quality indicator function indicates that theinput function is non-periodic. In some embodiments the variable gainmay be embodied as a non-linear function based on the quality indicatorfunction.

At 916, method 900 may determine whether the determined period is validbased on quality indicator. If the evaluation indicates a low quality at916, the previously determined period may be retained at 918. On theother hand, if the evaluation indicates a high quality at 916, theupdated frequency may generate an updated period at 920. In variousembodiments, the frequency measurement may be used and analyzed bymethod 900 to track the frequency during operation of a system.

Additional data may be received at 922, and method 900 may return to904. In various embodiments, method 900 may be performed immediatelyupon receipt of new data. In one specific embodiment, receipt of newdata may be tied to internal device processes such as the waveformsampling process, rather than in response to an external event (e.g.,zero crossing).

FIG. 10 illustrates a functional block diagram of a system 1000configured to estimate the period of a waveform consistent withembodiments of the present disclosure. System 1000 may be implementedusing hardware, software, firmware, and/or any combination thereof.Moreover, certain components or functions described herein may beassociated with other devices or performed by other devices. Thespecifically illustrated configuration is merely representative of oneembodiment consistent with the present disclosure. In certainembodiments, the system 1000 may comprise an IED. The specificallyillustrated configuration is merely representative of one embodimentconsistent with the present disclosure.

System 1000 may include a communications interface 1016 configured tocommunicate with other devices and/or systems. In certain embodiments,the communications interface 1016 may facilitate direct communicationwith another device or may communicate with one or more devices using acommunications network. Communications interface 1016 may be configuredfor communication using a variety of communication media and datacommunication protocols (e.g., Ethernet, IEC 61850, etc.).

System 1000 may further include a time subsystem 1012, which may be usedto receive a time signal (e.g., a common time reference) allowing system1000 to apply a time-stamp to the acquired samples. In variousembodiments, time subsystem 1012 may comprise a GNSS receiver, IRIG-Breceiver, a WWVB or WWV receiver and the like. In certain embodiments, acommon time reference may be received via communications interface 1016,and accordingly, a separate time input may not be required fortime-stamping and/or synchronization operations. One such embodiment mayemploy the IEEE 1588 protocol.

Time subsystem 1012 may further be configured to associate a time stampbased on the common time signal with representations of electricalconditions on the output of one or more electrical generators or one ormore electrical buses. Using such time-stamped representations, system1000 may determine whether a generator is synchronized to an electricalbus to which the generator is to be connected.

A monitored equipment interface 1008 may be configured to receive statusinformation from, and issue control instructions to, a piece ofmonitored equipment (such as a circuit breaker, recloser, etc.). Invarious embodiments monitored equipment interface 1008 may be incommunication with one or more breakers or re-closers that mayselectively connect or disconnect an electrical load. The monitoredinterface may be used in various embodiments to implement controlinstructions based on the frequency of a monitored waveform.

Processor 1024 may be configured to process communications received viacommunications interface 1016, time subsystem 1012, and/or monitoredequipment interface 1008. Processor 1024 may operate using any number ofprocessing rates and architectures. Processor 1024 may be configured toperform various algorithms and calculations described herein. Processor1024 may be embodied as a general purpose integrated circuit, anapplication specific integrated circuit, a field-programmable gatearray, and/or any other suitable programmable logic device.

In certain embodiments, system 1000 may include a sensor component 1010.In the illustrated embodiment, sensor component 1010 is configured togather data directly from equipment such as a conductor (not shown) andmay use, for example, transformers 1002 and 1014 and A/D converters 1018that may sample and/or digitize filtered waveforms to form correspondingdigitized current and voltage signals provided to data bus 1042. Current(I) and voltage (V) inputs may be secondary inputs from instrumenttransformers such as, CTs and VTs, connected to a generator output or anelectrical bus. A/D converters 1018 may include a single A/D converteror separate A/D converters for each incoming signal. A/D converters 1018may be connected to processor 1024 by way of data bus 1042, throughwhich digitized representations of current and voltage signals may betransmitted to processor 1024.

A period determination subsystem 1032 may be configured to analyze dataassociated with a waveform and determine a period of the waveform. Invarious embodiments consistent with the present disclosure, perioddetermination subsystem 1032 may solve Eq. 3 using various techniquesdisclosed herein. In other embodiments, period determination subsystem1032 may solve Eq. 1, or other functions from which the period of awaveform may be determined. In some embodiments, period determinationsubsystem 1032 may further be configured to implement a low pass filterto improve the accuracy of the determined waveform period.

A quality indicator subsystem 1034 may be configured to determine thequality of an estimated period determined by period determinationsubsystem 1032. In some embodiments, quality indicator subsystem 1034may assess the periodicity of the input waveform using Eq. 2 and varioustechniques disclosed herein. In some embodiments, the quality may bereflected as a numerical value indicating the periodicity of the inputsignal. In some embodiments, a value of 1 may be used to quantify acompletely periodic waveform. A value below, but close to 1 may reflecta nearly-periodic waveform, such as a frequency ramp, magnitude ramp, oroscillations in an electric power system. A value substantially below 1may quantify a non-periodic waveform. In some embodiments, a thresholdvalue may be established to differentiate between periodic andnon-periodic waveforms. In some embodiments, quality indicator subsystem1034 may further be configured to implement a low pass filter to improvethe accuracy of the quality indicator subsystem output.

An interpolation subsystem 1036 may be configured to interpolate thevalue of the period. In some embodiments, interpolation subsystem 1036may interpolate between successive samples in order to more accuratelyestimate the period of a waveform.

A estimated period adjustment subsystem 1038 may be configured toincrease or decrease an estimate of the period of the waveform. In someembodiments, period estimate adjustment subsystem may be incrementallyadjusted to determine the period of the waveform period. In embodimentsin which period determination subsystem 1032 implements Eq. 3A,estimated period adjustment subsystem 1038 may increase the estimatedperiod in a successive calculation if the result is positive, and maydecrease the estimated period in a successive calculation if the resultis negative.

A control action subsystem may be configured to implement a controlaction based on a determination of the period of a waveform. Changes inthe frequency of an alternating current associated with an electricpower distribution system may provide an indication of changes withinthe system. For example, where the system is overloaded, the frequencymay begin to decline. In such circumstances, system 1000 may detect thedeclining frequency and may reduce the load connected to the system byimplementing a control action. Shedding of loads is merely one exampleof a control action that may be implemented based on a change infrequency detected by system 1000.

While specific embodiments and applications of the disclosure have beenillustrated and described, it is to be understood that the disclosure isnot limited to the precise configurations and components disclosedherein. For example, the systems and methods described herein may beapplied to an industrial electric power delivery system or an electricpower delivery system implemented in a boat or oil platform that may notinclude long-distance transmission of high-voltage power. Moreover,principles described herein may also be utilized for protecting anelectric system from over-frequency conditions, wherein power generationwould be shed rather than load to reduce effects on the system.Accordingly, many changes may be made to the details of theabove-described embodiments without departing from the underlyingprinciples of this disclosure. The scope of the present inventionshould, therefore, be determined only by the following claims.

What is claimed is:
 1. A system configured to determine a period of a waveform, the system comprising: an input configured to receive a signal comprising a representation of the waveform; a period determination subsystem configured to calculate an estimated period of the signal based on a period determination function; an estimated period adjustment subsystem configured to determine an adjustment to the estimated period based on a result of the period determination function; a quality indicator subsystem configured to: evaluate a measurement quality indicator function based on the estimated period, and selectively update the period of the waveform based on the measurement quality indicator; and a control action subsystem configured to implement a control action based on the period of the waveform.
 2. The system of claim 1 wherein the period determination subsystem is further configured to incorporate the adjustment into a subsequent estimated period.
 3. The system of claim 3, wherein the period determination subsystem and the estimated period adjustment subsystem are further configured to determine a plurality of adjustments to the estimated period.
 4. The system of claim 1, wherein the waveform represents one of an alternating current and an alternating voltage in an electric power distribution system and the control action comprises an adjustment to a load associated with the electric power distribution system.
 5. The system of claim 1, wherein the period determination subsystem is configured to calculate the estimated period based on a time shift in the period determination function that results in a zero value of an integrated product of the input and the derivative of the input at the time shift.
 6. The system of claim 1, wherein the period determination function comprises an integral of the product over a time shift.
 7. The system of claim 1, wherein the period determination function is represented as: B(t,T)=∫_(t-T) ^(t) x(t−T)·x′(t)dt T ₀(t)=T for which B(t,T) is at zero
 8. The system of claim 7, wherein the estimated period adjustment subsystem is configured to increase the estimated period when B(t,T) is positive and to decrease the estimated period when B(t,T) is negative.
 9. The system of claim 1, further comprising an interpolating subsystem configured to interpolate between a plurality of adjacent samples associated with the input signal to determine the estimated period.
 10. The system of claim 1, wherein the measurement quality indictor function comprises: ${{A\left( {t,T} \right)} = \frac{\int_{t - T}^{t}{{{x(t)} \cdot {x\left( {t - T} \right)}}\ {t}}}{\int_{t - {2T}}^{t}{{{x(t)} \cdot {x(t)}}\ {t}}}},$ where T comprises the estimated period.
 11. The systems of claim 1, wherein the quality indicator subsystem is further configured to: establish a threshold associated with the measurement quality indicator; update the period of the waveform when the measurement quality indicator exceeds the threshold; retain a previously determined period of the waveform when the measurement quality indicator is below the threshold.
 12. The systems of claim 1, wherein the quality indicator subsystem is further configured to establish a variable gain to be applied to the signal comprising the representation of the waveform, the variable gain being proportional to the measurement quality indicator function.
 13. A method for estimating a period of a waveform, the method comprising: receiving a signal comprising a representation of the waveform; calculating an estimated period of the signal based on a period determination function; determining an adjustment to the estimated period based on a result of the period determination function; evaluating a measurement quality indictor function based on the estimated period; selectively updating the period of the input signal based on the measurement quality indicator; and performing a control action based on the period of the waveform.
 14. The method of claim 13 further incorporating the adjustment into a subsequent estimated period.
 15. The method of claim 14, further comprising identifying the period by adding a plurality of adjustments to the estimated period.
 16. The method of claim 13, wherein waveform represents an alternating current in an electric power distribution system and the control action comprises adjusting a load associated with the electric power distribution system.
 17. The method of claim 13, wherein calculating the estimated period based on the period determination function comprises: calculating a time shift that results in a zero value of an integrated product of the waveform and the derivative of the waveform at the time shift.
 18. The method of claim 13, wherein the period determination function comprises an integral of the product over a time shift.
 19. The method of claim 13, wherein the period determination function is represented as: B(t,T)=∫_(t-T) ^(t) x(t−T)·x′(t)dt T ₀(t)=T for which B(t,T) is at zero
 20. The method of claim 19, wherein the adjustment to the estimated period comprises: increasing the estimated period when B(t,T) is positive; and decreasing the estimated period when B(t,T) is negative.
 21. The method of claim 13, further comprising interpolating between a plurality of adjacent samples associated with the representation of the input signal to determine the estimated period.
 22. The method of claim 13, wherein the measurement quality indictor function comprises: ${{A\left( {t,T} \right)} = \frac{\int_{t - T}^{t}{{{x(t)} \cdot {x\left( {t - T} \right)}}\ {t}}}{\int_{t - {2T}}^{t}{{{x(t)} \cdot {x(t)}}\ {t}}}},$ where T comprises the estimated period.
 23. The method of claim 13, further comprising: establishing a variable gain based on the measurement quality indicator function; and applying the variable gain the signal representing the waveform.
 24. The method of claim 13, further comprising: establishing a threshold associated with the measurement quality indicator; updating the period of the waveform when the measurement quality indicator exceeds the threshold; retaining a previously determined period of the waveform when the measurement quality indicator is below the threshold.
 25. The method of claim 13, further comprising implementing a low pass filter on the period determination function. 